Heat transport by turbulent Rayleigh–Bénard convection forPr≃ 0.8 and 4 × 10¹¹≲Ra≲ 2 × 10¹⁴: ultimate-state transition for aspect ratio Γ = 1.00

Abstract

We report experimental results for heat-transport measurements, in the form of the Nusselt number N u, by turbulent Rayleigh-Bénard convection (RBC) in a cylindrical sample of aspect ratio ≡ D/L = 1.00 (D = 1.12 m is the diameter and L = 1.12 m the height) and compare them with previously reported results for = 0.50. The measurements were made using sulfur hexafluoride at pressures up to 19 bars as the fluid. They are for the Rayleighnumber range 4 × 10 11 Ra 2 × 10 14 and for Prandtl numbers Pr between 0.79 and 0.86. For Ra < Ra * 1 2 × 10 13 we find N u = N 0 Ra γ eff with γ eff = 0.321 ± 0.002 and N 0 = 0.0776, consistent with classical turbulent RBC in a system with laminar boundary layers (BLs) below the top and above the bottom 6 International Collaboration for Turbulence Research. 2 plate and with the prediction of Grossmann and Lohse. For Ra > Ra * 1 the data rise above the classical-state power-law and show greater scatter. In analogy to similar behavior observed for = 0.50, we interpret this observation as the onset of the transition to the ultimate state. Within our resolution this onset occurs at nearly the same value of Ra * 1 as it does for = 0.50. This differs from an earlier estimate by Roche et al (2010 New J. Phys. 12 085014), which yielded a transition at Ra U 1.3 × 10 11 −2.5±0.5 . A -independent Ra * 1 would suggest that the BL shear transition is induced by fluctuations on a scale less than the sample dimensions rather than by a global -dependent flow mode. Within the resolution of the measurements the heat transport above Ra * 1 is equal for the two values, suggesting a universal aspect of the ultimate-state transition and properties. The enhanced scatter of N u in the transition region, which exceeds the experimental resolution, indicates an intrinsic irreproducibility of the state of the system. Several previous measurements for = 1.00 are re-examined and compared with the present results. None of them identified the ultimate-state transition.

Has the ultimate state of turbulent thermal convection been observed?

“…Our analysis strongly suggests that the present claims (Chavanne et al 1997(Chavanne et al , 2001Roche et al 2010;He et al 2012aHe et al ,b, 2014) are most likely not justified, leaving open an important consideration of relating them to non-OberbeckBoussinesq effects, and the very important and intriguing question of the transition (if it occurs) to the ultimate state of Rayleigh-Bénard convection remains open.…”

“…and in the SF 6 Göttingen experiments at ambient temperature He et al 2012aHe et al ,b, 2014. This paper extends our recent experimental studies considerably (Urban, Musilová & Skrbek 2011;Urban et al 2012Urban et al , 2014, where we discussed in detail issues such as the influence of the shape (aspect ratio) and material properties of the experimental cells, various corrections to the raw data and advantages of using a clean cryogenic environment.…”

Urbanet al.Reply:

“…Based on the analysis of Grossmann and Lohse [10], He et al consider as a more realistic value Ra à ' 10 14 , but neglect to mention that [10] requires four parameters that are extracted by fitting selected (not all, e.g., [6]) high Ra experiments, performed using various working fluids in cells of various size and À, and its predictive capability is necessarily limited. Figure 1 compares selected compensated Nu(Ra) data with the two sets of À ¼ 1=2 data of He et al for their closed SF 6 sample [7,9] corresponding to different T m -T U , demonstrating clearly that the temperature outside their cell, T U , influences the observed Nu ¼ NuðRaÞ dependence (compare also with [11], claiming no transition to an ultimate regime for 10 9 & Ra & 3  10 14 ). Figure 1 shows clear departures to steeper Nu(Ra) scaling, indicative of transition to the ultimate regime, observed in high Ra experiments for 10 11 & Ra à & 10 15 ; other nominally identical experiments (such as [4], not shown) do not display them.…”

“…We agree with He et al that if the working fluid sample were strictly Boussinesq, then obviously T m ¼ T c . Experimentally, however, this is never the case, especially if the working fluid is used close to its critical point [4][5][6], and we appreciate that He et al show in [7] as well as in their Comment [8] that this effect (small, but increasing with Ra, it might be interesting to evaluate it for the six data points above Ra à 2 ' 5  10 14 in the aspect ratio À ¼ 1=2 cell [9]) is present for their own data. To justify that the sample can be treated as Boussinesq, various phenomenological requirements are usually applied, such as ðT b À T t Þ & 0:2, where is the thermal expansion coefficient.…”

Highly turbulent Taylor-Couette flow : direct numerical simulations

“…For RB flow, the transition to an ultimate regime was first qualitatively predicted by Kraichnan [74], and later quantitatively by Grossmann & Lohse [70,78,154] and then experimentally found by several groups [75,76,169,170]. It lies outside the present reach of DNS.…”